Ch08 Sampling Methods and the Central Limit Theorem.docx

Chapter 8 Sampling Methods and the Central Limit Theorem True/False 1. In stratified random sampling, a population is divided into strata using naturally occurring geographic or other boundaries. Then strata are randomly selected and a random sample is collected from each strata. Answer: False 2. In cluster sampling, a population is divided into clusters using naturally occurring geographic or other boundaries. Then clusters are randomly selected and a random sample is collected from each cluster. Answer: True 3. Sampling a population is often necessary because the cost of studying all the items in the population is prohibitive. Answer: True 4. It is often not feasible to study the entire population because it is impossible to observe all the items in the population. Answer: True 5. A simple random sample assumes that each item or person in the population has an equal chance of being included. Answer: True 6. The standard error of the mean is also be called sampling error. Answer: False 7. When systematic random sampling is used, the central limit theorem cannot be applied. Answer: False 8. When using stratified random sampling, the sampling error will be zero. Answer: False 9. In cluster sampling, a population is divided into subgroups called clusters and a sample is randomly selected from each cluster. Answer: False 10. In stratified random sampling, a population is divided into subgroups called strata and a sample is randomly selected from each stratum. Answer: True 11. When systematic random sampling is used, the central limit theorem cannot be applied. Answer: False 12. When using systematic random sampling, the sampling error will be zero. Answer: False 13. If probability sampling is done, each item in the population has a chance of being chosen. Answer: True 14. The items or individuals of the population are arranged in a file drawer alphabetically by date received. A random starting point is selected and then every kth member of the population is selected for the sample. This sampling method is called simple random sampling. Answer: False 15. If the size of a sample equals the size of the population, we would not expect any error in estimating the population parameter. Answer: True 16. We can expect some difference between sample statistics and the corresponding population parameters. This difference is called the sampling error. Answer: True 17. A sampling distribution of the means is a probability distribution consisting of a list of all possible sample means of a given sample size selected from a population and the probability of occurrence associated with each sample mean. Answer: True 18. The central limit theorem implies that sampling with an adequate sample size provides good estimates of population parameters. Answer: True 19. The central limit theorem implies that samples of size one or two are adequate to estimate population parameters. Answer: False 20. If a population is not normally distributed, the sampling distribution of the sample means tends to approximate a normal distribution. Answer: True 21. The Central Limit Theorem states that for a sufficiently large sample the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed with the mean of the sampling distribution equal to ?2 and the variance equal to ?2/n. Answer: False 22. The Central Limit Theorem states that if the sample size, n, is sufficiently large, the sampling distribution of the means will be approximately normal no matter whether the population is normally distributed, skewed, or uniform. Answer: True 23. Fish and game wardens estimate the average weight of the fish or game population by using creel checks and other devices. Based on this sample data, a warden might estimate that the mean weight of Coho salmon caught in Lake Michigan is 2.5 pounds. This single number is called a point estimate of the unknown population parameter. Answer: True 24. Based on the sampling distribution of the means and the central limit theorem, the sample mean can be used as a good estimator of the population mean, assuming that the size of the sample is sufficiently large. Answer: True 25. If 40 samples of size 21 were selected from a population of 22,493, we would expect the mean of the sample means and the population mean to be close but not exactly equal. Answer: True 26. An estimate of the population mean based on a large sample is less reliable than an estimate made using a small sample. Answer: False 27. If the sample size keeps getting larger and larger and finally equals the size of the population, there would be no error in predicting the population mean because the sample size and the size of the population would be the same. Answer: True 28. The standard error of the mean will vary according to the size of the sample. As the sample size n gets larger, the variability of the sample means gets smaller. Answer: True 29. To determine the value of the standard error of the mean, the total error is divided by the sample size. Answer: False Multiple Choice 30. What is it called when all the items in a population have a chance of being selected in a sample? A) Random sampling B) z-score C) Sampling error D) Nonprobability sampling Answer: A 31. What is the difference between a sample mean and the population mean called? A) Standard error of the mean B) Sampling error C) Interval estimate D) Point estimate Answer: B 32. What sample statistic is used to estimate a population parameter? A) Parameter B) Sampling error C) Point estimate D) Interval estimate Answer: C 33. Suppose we select every fifth invoice in a file. What type of sampling is this? A) Random B) Cluster C) Stratified D) Systematic Answer: D 34. All possible samples of size n are selected from a population and the mean of each sample is determined. What is the mean of the sample means? A) Exactly the same as the population mean B) Larger than the population mean C) Smaller than the population mean D) Cannot be estimated in advance Answer: A 35. When dividing a population into subgroups so that a random sample from each subgroup can be collected, what type of sampling is used? A) simple random sampling B) systematic sampling C) stratified random sampling D) cluster sampling Answer: C 36. The true sampling error is usually not known because A) a sample is collected from a population B) µ is a random variable C) ?2 is unknown D) The sample mean cannot be computed Answer: A 37. Based on the central limit theorem, the size of the sampling error is A) Directly related to the sample size, i.e., the larger the sample size the larger the sampling error. B) Directly related to the population mean, i.e., the larger the mean, the larger the sampling error C) Inversely related to the sample size, i.e., the larger the sample size the smaller the sampling error. D) Inversely related to the population standard deviation, i.e., the smaller the standard deviation, the larger the sampling error. Answer: C 38. For a distribution of sample means constructed by sampling 5 items from a population of 15, A) the sample size is 15 B) there will be 3003 possible sample means C) the mean of the sample means will be 3 D) the standard error will be 1 Answer: B 39. As the size of the sample increases, what happens to the shape of the sampling means? A) Cannot be predicted in advance B) Approaches a normal distribution C) Positively skewed D) Negatively skewed Answer: B 40. Manufacturers were subdivided into groups by volume of sales. Those with more than $100 million in sales were classified as Class A large; those from $50 to $100 million as Class A medium size; and those between $25 and $50 million..., and so on. Samples were then selected from each of these groups. What is this type of sampling called? A) Simple random B) Stratified random C) Cluster D) Systematic Answer: B 41. An experiment involves selecting a random sample of 256 middle managers at random for study. One item of interest is their mean annual income. The sample mean is computed to be $35,420 and the sample standard deviation is $2,050. What is the standard error of the mean? A) $128.125 B) $138.36 C) $2,050 D) $8.01 Answer: A 42. The wildlife department has been feeding a special food to rainbow trout fingerlings in a pond. A sample of the weights of 40 trout revealed that the mean weight is 402.7 grams and the standard deviation 8.8 grams. What is the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams? A) 0.3783 B) 0.0228 C) 1.0 D) 0.5 Answer: B 43. Suppose a research firm conducted a survey to determine the average amount of money steady smokers spend on cigarettes during a week. A sample of 100 steady smokers revealed that the sample mean is $20 and the sample standard deviation is $5. What is the probability that a sample of 100 steady smokers spend between $19 and $21? A) 0.4772 B) 0.0228 C) 0.9544 D) $20 Answer: C 44. The mean weight of trucks traveling on a particular section of I-475 is not known. A state highway inspector needs an estimate of the mean. He selects a random sample of 49 trucks passing the weighing station and finds the mean is 15.8 tons, with a standard deviation of the sample of 4.2 tons. What is probability that a truck will weigh less than 14.3 tons? A) 0.0062 B) 0.3632 C) 0.1368 D) 0.4938 Answer: A 45. A statewide sample survey is to be made. First, the state is subdivided into counties. Seven counties are selected at random and further sampling is concentrated on these seven counties. What type of sampling is this? A) Simple random B) Nonproportional C) Cluster D) Stratified Answer: C 46. Which of the following is the standard error of the mean? A) ? B) x/n C) D) s Answer: C 47. Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The average tread wear was found to be 50,000 miles with a standard deviation of 3,500 miles. What is the best estimate of the average tread life in miles for the entire population of these tires? A) 50,000 B) 3,500 C) (50,000/100) D) (3,500/100) Answer: A 48. The mean of all possible sample means is equal to the A) population variance. B) . C) sample variance. D) population mean. Answer: D 49. Sampling error is the difference between a corresponding sample statistic and the A) sample mean. B) biased sample. C) population parameter. D) chance error. Answer: C 50. For a population that is not normally distributed, the distribution of the sample means will A) be negatively skewed. B) approach the normal distribution. C) be positively skewed. D) take the same shape as the population. Answer: B Fill-in-the-Blank 51. Sampling from a population may be preferred because studying all the items in a population may be too _____________. Answer: costly 52. Sampling from a population may be preferred because, contacting or listing the entire population would be too ___________ consuming. Answer: time 53. Sampling from a population may be preferred because, sometimes collecting statistics requires that products be ___________. Answer: destroyed 54. How many different samples of size 3 can be collected from a population of ten items? ___________ Answer: 120 55. What type of sampling is it when a population is first divided into subgroups and then a sample is selected from each subgroup? _________ Answer: stratified random sampling 56. Auditors may select every 20th file starting with say, the 5th file in the top drawer. Then file numbers 25, 45, 65, 85, . . . are audited. What type of sampling is this? _________________ Answer: systematic 57. What will the sampling distribution of the mean be if a population is normally distributed? ___________________________ Answer: normally distributed 58. What is the name of the standard deviation of the sampling distribution of the sample means called? ______________________ Answer: standard error of the mean 59. The process of making statements about a population based on a sample from the population is called ______________________ Answer: statistical inference 60. What is a characteristic of a population called? _______________ Answer: parameter 61. What is the relationship of the standard deviation of the sampling distribution of the mean to the standard deviation of the population under study? ___________ Answer: smaller 62. For a sampling distribution of the means, what percent of the means would be between ± 1.96 standard deviations? ________________________ Answer: 95% 63. As the sample size (n) increases, what happens to the spread or dispersion of the distribution of the sample means? ___________ Answer: decreases 64. If the sample size equals the population size, what is the sampling error? _________ Answer: zero 65. For populations scattered in a wide area, what is the preferred technique for sampling? ____________ Answer: cluster sampling 66. Which sampling method would be best to use if the population can be divided into homogeneous subgroups? ______________________ Answer: stratified random sampling 67. Which sampling method would you use if every k-th item in the population sequence is selected? ______________________ Answer: systematic random sampling Multiple Choice Use the following to answer questions 68-71: An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes. 68. What assumptions do you need to make about the shape of the population distribution of all possible tax preparation times to make inferences about the average time to complete a tax form? A) The population distribution is skewed to the right. B) The population distribution is skewed to the left. C) The population distribution is normal. D) The shape of the population distribution does not matter. Answer: D 69. What is the standard error of the mean? A) 14 minutes B) 140 minutes C) 1.4 minutes D) 90 minutes Answer: A 70. What is the probability that the mean completion time will be more than 120 minutes? A) Approximately zero B) 0.0832 C) 0.4168 D) 0.0162 Answer: D 71. What is the probability that the mean completion time is between 1 and 2 hours, i.e., 60 and 120 minutes? A) Approximately 1. B) 0.1664 C) 0.8336 D) 0.9676 Answer: D Use the following to answer questions 72-75: A group of statistics students decided to conduct a survey at their university to find the average (mean) amount of time students spend studying per week. Based on a simple random sample, they surveyed 144 students. The statistics showed that students studied an average of 20 hours per week with a standard deviation of 10 hours. 72. What is the standard error of the mean? A) 0.83 B) 10 C) 0.5 D) 2 Answer: A 73. What is the probability that a sample mean would exceed 20 hours per week? A) 1.0 B) 0.5 C) 1.96 D) Cannot be calculated based on the given information. Answer: B 74. What is the probability of finding a sample mean less than 18 hours? A) 0.4820 B) 0.4920 C) 0.0080 D) 0.0180 Answer: C 75. What is the probability that average student study time is between 18 and 22 hours? A) 0.9640 B) 0.0160 C) 0.0360 D) 0.9840 Answer: D Use the following to answer questions 76-79: The Intelligence Quotient (IQ) test scores are normally distributed with a mean of 100 and a standard deviation of 15. 76. What is the probability that a person would score 130 or more on the test? A) 0.0228 B) 0.9772 C) 0.4772 D) 0.9544 Answer: A 77. What is the probability that a person would score between 85 and 115? A) 0.3413 B) 0.6826 C) 1.00 D) very likely Answer: B 78. You enrolled in a class of 25 students. What is the probability that the class' average IQ exceeds 130? A) 0.0228 B) 0.9772 C) Approximately zero D) 0.9544 Answer: C 79. Given a class with 25 students, what is the probability that the class' average IQ score is between 85 and 115? A) 0.4987 B) 3.00 C) 0.0026 D) 0.9974 Answer: D Use the following to answer questions 80-83: Truck tire life is normally distributed with a mean of 60,000 miles and a standard deviation of 4,000 miles. 80. What is the probability that a tire will last 72,000 miles or more? A) 0.0013 B) 0.9987 C) 0.4987 D) 0.9544 Answer: A 81. What is the probability that a tire lasts between 54,000 and 66,000 miles? A) 0.4332 B) 0.8664 C) 1.00 D) very likely Answer: B 82. You bought four tires. What is the probability that the average mileage of the four tires exceeds 66,000 miles? A) 0.0013 B) 0.9987 C) 0.4987 D) 0.9544 Answer: A 83. For a set of four tires, what is the probability that the average tire life is between 57,000 and 63,000 miles? A) 0.4332 B) 0.8664 C) 1.00 D) very likely Answer: B Essay 84. In practice, researchers collect a sample of size “n” and compute point estimates of population parameters. Based on the Central Limit Theorem, how can sampling error be reduced? Answer: Increasing the sample size; 85. What are the two population parameters that specify the location and dispersion for a normal distribution? Answer: or the population mean and or the population variance 86. What are the two population parameters that specify the location and dispersion for the sampling distribution of the mean? Answer: or the population mean and or the population standard error 87. How is the Central Limit Theorem helpful when computing probabilities? Answer: The Central Limit Theorem allows the use of the normal distribution and the standard normal z to compute probabilities.